Most of the literature on mathematical models of disease dynamics treats deterministic models of the type of (REF ). When an epidemics is established, and each compartment of the model contains a significant proportion of the total population, if \(N\)  is large enough, the ODE (REF ) is a good model for the epidemics. The original stochastic model (REF ), which we believe to be more realistic than the (REF ), can be considered as a stochastic perturbation of (REF ). However, we know from the work of [1]}, that small Brownian perturbations of an ODE will eventually produce a large deviation from its law of large numbers limit. For instance, if the ODE starts in the basin of attraction of an locally stable equilibrium,
the solution of the ODE converges to that equilibrium, and stays for ever close to that equilibrium. The theory of Freidlin and Wentzell, based upon the theory of Large Deviations, predicts that soon or later the solution of a random perturbation of that ODE will exit the basin of attraction of the equilibrium. The aim of this paper is to show that the Poissonian perturbation (REF ) of (REF ) behaves similarly. This should allow us to predicts the time taken by an endemic equilibrium to cease, and a disease–free equilibrium to replace it.
